Decay rates of solutions to a von Kármán system for viscoelastic plates with memory

Muñoz Rivera, Jaime E.; Perla Menzala, Gustavo

doi:10.1090/qam/1672191

Authors: Jaime E. Muñoz Rivera and Gustavo Perla Menzala
Journal: Quart. Appl. Math. 57 (1999), 181-200
MSC: Primary 74H45; Secondary 35Q72, 45K05, 74D10, 74K20
DOI: https://doi.org/10.1090/qam/1672191
MathSciNet review: MR1672191
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Abstract: We consider the dynamical von Kármán equations for viscoelastic plates under the presence of a long-range memory. We find uniform rates of decay (in time) of the energy, provided that suitable assumptions on the relaxation functions are given. Namely, if the relaxation decays exponentially, then the first-order energy also decays exponentially, When the relaxation $g$ satisfies \[ - {c_1}{g^{1 + \frac {1}{p}}}\left ( t \right ) \le g’\left ( t \right ) \le - {c_0}g{\left ( t \right )^{1 + \frac {1}{p}}}, \qquad 0 \le g”\left ( t \right ) \le {c_2}{g^{1 + \frac {1}{p}}}\left ( t \right ) , \: \textrm {and} \] \[ g, {g^{1 + \frac {1}{p}}} \in {L^1}\left ( \mathbb {R} \right ) \textrm {with} \: p > 2 ,\] then the energy decays as $\frac {1}{\left ( 1 +t \right )^{p}}$. A new Liapunov functional is built for this problem.

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